Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{5z^3 - 125z}{7z^2 + 105z + 350}$
First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {5z(z^2 - 25)} {7(z^2 + 15z + 50)} $ $ p = \dfrac{5z}{7} \cdot \dfrac{z^2 - 25}{z^2 + 15z + 50} $ Next factor the numerator and denominator. $ p = \dfrac{5z}{7} \cdot \dfrac{(z + 5)(z - 5)}{(z + 5)(z + 10)}$ Assuming $z \neq -5$ , we can cancel the $z + 5$ $ p = \dfrac{5z}{7} \cdot \dfrac{z - 5}{z + 10}$ Therefore: $ p = \dfrac{ 5z(z - 5)}{ 7(z + 10)}$, $z \neq -5$